Again here as in the question of the why the factorial of zero is one there is no real explanation of Why. What we are offered is a reason based on the mathematical systems definition. But does this really tell us why. Again I offer the book Negative Math: How Mathematical Rules Can Be Positively Bent: Alberto A. Martínez: 9780691123097: Amazon.com: Books which talks about alternative mathematical systems that we could have chosen, and how they may be in better alignment with the world but not as systematic, and so we chose a mathematical system that was as systematic as possible given the nature of numbers rather than one that fits what we experience in the world and this means we are always having to bridge the gap between what math tells us and what we experience. The fact that we chose a mathematics that does not fit the world completely because we wanted it to be as systematic as possible is why we call our mathematics Platonic. It also explains probably why it took so long for our mathematics to reach maturity. There was resistance all the way to Zero, to Irrational Numbers, to Imaginary Numbers, etc. to all the things that were counter intuitive about the system of mathematics.

So I propose that the real reason why is that we choose a mathematics that was as systematic as possible given the nature of numbers, and that the fact that Factorial of Zero is One, and Zeroth Power of any Number is One are two outcomes of this systematicity of the system of Mathematics that we chose. Another outcome is that n/0 is undefined. Another is that -1 = e^pi*i and the number series equals -1/12th. In other words there are anomalies in the system of mathematics we chose which are inexplicable.

This is why we can say that Mathematics is scientific. It accommodates itself to the nature of number systems and their systematicity in themselves more than it accommodates itself to us. It discovers anomalous characteristics in numbers and other types of orderings that mathematics deals with that changes our view of what number or order is fundamentally. But this means we continually have to translate back and forth between the realm of number to our everyday experience of the world. And the reason it took so long to develop modern mathematics is that we had to learn to put ourselves aside and give the phenomena of number the last say on its own nature as something other than we might presuppose. So we get paradigm shifts and episteme shifts in the history of mathematics just like any other science, and just like any other science it changes the way we see the world when we allow the phenomena to be itself beyond our projections on it.